Sketch the graph of $y=\left| b+\frac{a}{x+b} \right|$, where $0

Question

Sketch the graph of $y=\left| b+\frac{a}{x+b} \right|$, where $0 < a < b$, stating the equations of any asymptotes and the coordinates of the points that cross the axes.[3] Hence solve the inequality $-x\le \left| b+\frac{a}{x+b} \right|$.[2] Using part (ii), given that $a=1$, $b=2$, solve the inequality ${{\mathrm{e}}^{x}}\le \left| b+\frac{a}{b-{{\mathrm{e}}^{x}}} \right|$.[2]

Accepted Answer

(i) Asymptotes: $x=-b$, $y=b$; intercepts: $\left(-b-\dfrac ab,0\right)$, $\left(0,b+\dfrac ab\right)$ (ii) $x\ge-\sqrt{a+b^2}$, $x\ne-b$ (iii) $x\le\ln\sqrt5$, $x\ne\ln2$

2023 ACJC Promo Q2

2023 ACJC Promo Q2

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